OCT Imaging in Glaucoma: A guide for practitioners

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021

K. H. Park, T.-W. Kim (eds.)OCT Imaging in Glaucomahttps://doi.org/10.1007/978-981-16-1178-0_1

Principles of OCT Imaging

Kyoung Min Lee¹  

(1)

Seoul National University Boramae Medical Center, Dongjak-gu, Seoul, Korea

Kyoung Min Lee

Email: [email protected]

Abstract

Optical coherence tomography (OCT) uses the interference between the light waves reflected by the reference and sample arms to obtain spatial information on tissue microstructure, which is used to construct an in-vivo cross-sectional image. OCT has become an essential part of daily practice in the field of ophthalmology and glaucoma over the past 30 years. This success was possible by tremendous advances of its technology toward better sensitivity and faster scan speed from Time-domain to Spectral-domain and Swept-source OCT. This chapter was written to describe the basic principles of OCT and how they enable its various applications, which are crucial to understand how such advances of OCT have been possible in the past, and why OCT still has boundless potential for future growth.

Keywords

Optical coherence tomographyInterferometerFourier transformationSpectral-domain OCTSwept-source OCT

1 Introduction

Modern progress in medicine has largely been driven by the advent of tomographic imaging techniques such as X-ray computed tomography (CT), magnetic resonance imaging (MRI), and ultrasound imaging. These can generate a cross-sectional image of living tissue, thereby locating the diseased portion. Optical coherence tomography (OCT), meanwhile, can be described as the optical analog of ultrasound imaging (Huang et al. 1991; Fercher et al. 2003; Fercher 2010; Popescu et al. 2011; Drexler et al. 2014; Fujimoto and Swanson 2016; Aumann et al. 2019). In ultrasound imaging, a sound pulse is launched, the echoes of which are measured to create a cross-sectional image. In OCT, correspondingly, a light pulse is launched, but its reflections—in the aspect of its delays—cannot be measured directly because of its fast velocity; instead, the interference between the light waves reflected by the reference and sample arms, respectively, is measured to obtain spatial information on tissue microstructure. To understand this more deeply, let us start with light and its wave-like behavior.

2 Light

Light is an electromagnetic phenomenon. A particle that carries an electric charge (q0) will generate an electric field (E) with different amounts of force (Fe) on a charge at different locations:

. This in turn will generate an electromagnetic field that is well explained by Maxwell’s equations (Brezinski 2006a) as follows:

1.

Gauss’ law for electric fields: ∇ ∙ D = ρ, which means that the net outward flow (D) through a closed surface is equal to the total charge (ρ) enclosed by that surface.

2.

Gauss’ law for a magnetic field: ∇ ∙ B = 0, which means that the divergence of the magnetic flux (B) is always zero and that there are no isolated magnetic changes.

3.

Faraday’s law:

, which means that a magnetic field (B) change over time (t) produces an electric field (E).

4.

Ampere’s law:

, which means that a changing electric field (D) produces a magnetic field (H).

That is, a changing electric field produces a magnetic field, whereas a changing magnetic field produces an electric field. Since the two fields recreate each other indefinitely, a light can propagate in free space as an oscillating wave with the electric and magnetic fields located perpendicularly to each other (Fig. 1a). Each light, as an oscillating wave, has a frequency (f) and wavelength (λ): c = f × λ. The energy transmitted by the light is proportional to the frequency: E = h × f (h =Planck’s constant).

Fig. 1

Interference and Michelson interferometer. By mutual production, electric and magnetic waves can propagate eternally in space with a fixed oscillatory component: a frequency (a; figure taken from Wikipedia). The summation of two waves with the same frequency (b) can produce constructive (c) or destructive (d) interference according to a phase delay. The interference can be measured using the Michelson interferometer (e). The pattern of interference is determined by twice the length difference between M1 and M2 (round trip)

2.1 Interference

A point in a wave is determined by its spatial location (x) and time (t) (Fig. 1b).

W1(x, t) = A cos (kx − ωt), where A is the peak amplitude, $$ k=\raisebox{1ex}{$2\pi $}\!\left/ \!\raisebox{-1ex}{$\lambda $}\right. $$ is the wavenumber, and ω = 2πf is the angular frequency of the wave. A second wave with a phase delay (φ) is W2(x, t) = A cos (kx − ωt + φ).

The summation of the two waves is as follows:

This equation reveals that the new wave shares the same frequency and wavelength (both k and ω are unchanged). The amplitude of the new wave is determined by a factor of $$ 2\cos \frac{\varphi }{2} $$ . When φ is an even multiple of π,

, and so the amplitude is doubled, which phenomenon is known as constructive interference (Fig. 1c). When φ is an odd multiple of π,

, and so the amplitude becomes zero, which phenomenon is known as destructive interference (Fig. 1d) (Brezinski 2006b).

More generally, if both the initial wave (W1) and the summation of the initial and reflected waves (W1 + W2) are provided, the phase delay (φ) of W2 can be calculated. The Michelson Interferometer is designed to measure this interference phenomenon (Fig. 1e). A beam splitter directs the course of light against two different mirrors (M1 and M2). Both mirrors reflect the lights: they share the same frequency and wavelength, but the lengths of their light paths are different (Fig. 1e). If the two mirrors reflect the lights equally, the length of difference alone determines the phase delay.

2.2 Coherence

To this point, calculations were based on light with a single frequency (monochromatic light). But real light sources are never monochromatic, even laser sources being quasi-monochromatic, for two reasons. First, light is produced by atomic transitions from a higher-energy state to a lower one, and the energy is directly related to the frequency: E = h × f. The problem is that while the atoms are in the excited state, they are colliding, which results in the loss or gain of some energy. So, the energy and the frequency vary, they are not fixed, even from the beginning. Second, the atoms emitting light are moving in different directions, which fact, given that the interaction of light with a moving object results in a Doppler shift, induces a frequency shift (Brezinski 2006b).

Coherence is a measure of inter-wave correlation. Since a wave is dependent on spatial location (x) and time (t), coherence is measured in both the spatial and temporal aspects. Here, let us focus on temporal coherence. As the frequencies of different waves get closer and closer, the time during which the waves are in unison–the coherence time–will become longer and longer. On the other hand, the lights have a broader spectrum of frequencies, the lower the chances are that the waves will be in unison, and, correspondingly, the shorter the coherence time becomes. The coherence length is the distance that light travels during the coherence time; it signifies that the properties of the beam retain relatively constant characteristics only over that unit of length. Using the Michelson interferometer, the length difference can be detected only within the limit of the coherence length. Therefore, the coherence length is directly associated with the axial resolution, and that is why low-coherence light, which has broader spectral bandwidth, is preferred for Time-domain OCT.

2.3 Diffraction

Diffraction refers to various phenomena that occur when a wave encounters an obstacle or a slit, the aperture effectively becoming a secondary source of the propagating wave. Therefore, even a well-focused, aberration-free converging lens never focuses light to a single point but always has a trace of diffraction (Brezinski 2006b). Diffraction of a circular aperture produces the bright central irradiance surrounded by the series of concentric rings: the former is known as the Airy spot which contains 84% of the total irradiance, and the latter is known as the Airy pattern. Since the system can only resolve the structures to the width of the Airy function, the diffraction determines the resolution limit of the system. It is particularly important for the lateral resolution which is essentially dependent on the optics of the imaging device. A larger diameter lens and shorter wavelengths result in smaller Airy spot and therefore higher resolutions (Brezinski 2006b).

2.4 Tissue Interactions

When light is within materials rather than in a vacuum, it interacts with atoms and molecules, and this interaction is best described as the dipole moment: separation of opposite charges (Brezinski 2006c). Then, the electrical dipole oscillates and generates a second propagating electromagnetic wave with a different velocity. This velocity change in the light is known as refractive index: $$ n=\raisebox{1ex}{$c$}\!\left/ \!\raisebox{-1ex}{$v$}\right. $$ .

The refractive index is frequency dependent: in a medium, different wavelengths will travel at different speeds, and this fact is called dispersion. Although dispersion can be used to decompose the light into its different frequency components as in Spectral-domain OCT, dispersion generally needs to be compensated for in OCT systems because dispersion makes the different wavelengths return at slightly different times.

When the impinging electromagnetic waves are near the resonance frequency, the electron absorbs the energy of a photon of a specific frequency and goes to a higher-energy state, a phenomenon known as absorption. More generally, atoms that are exposed to light absorb light energy and re-emit light in different directions with differing intensities, a phenomenon known as scattering. Keep in mind, meanwhile, that refractive index, dispersion, absorption, and scattering are related to each other in that they are all frequency (f) dependent.

3 Time-Domain OCT

The basic principle of Time-domain OCT is that of the Michelson interferometer’s use of the light of a low-coherence source (Fig. 2a) (Huang et al. 1991; Fercher et al. 2003; Fercher 2010; Popescu et al. 2011; Drexler et al. 2014; Fujimoto and Swanson 2016; Aumann et al. 2019; Brezinski 2006d). By replacing one of the mirrors with a sample, one can measure the interference of back-scattered light from the sample and the reflected light from the reference mirror. The back-reflected lights from the two arms (reference and sample) are combined and interfere only if the optical path lengths match within the coherence length. Interference fringe bursts, roughly the amplitudes of the interference, are detected by the photodiode. For each sample point, the reference mirror is scanned in the depth (z) direction, and a complete depth profile is generated at the beam position: this is the A-scan (amplitude scan). For a cross-sectional image, the scan beam is moved laterally across the line, and repeats A-scans in the same way: this is the B-scan (named after brightness scan in ultrasonography) (Fig. 2b) (Fercher et al. 2003; Fercher 2010; Popescu et al. 2011; Drexler et al. 2014; Fujimoto and Swanson 2016; Aumann et al. 2019).

Fig. 2

Working principle of Time-domain OCT (TD-OCT). For the axial image, a light from the light source is split into the reference beam and the central beam. Back-reflected light from both arms is combined again and recorded by the detector. To record one depth profile of the sample (an A-scan), the reference arm needs to be scanned (a). This has to be repeated for each lateral scan position to construct a volume scan (b). Therefore, the lateral resolution of OCT depends on the focusing ability of the probing beam: which is to say, the optics of the system. Figures reprinted from (Aumann et al. 2019)

To compute the axial resolution, the wavelength profile of a light source must be considered, because it determines the coherence length. Let us assume a Gaussian spectrum of wavelengths, then the wavelength profile is determined by its wavelength (λ0) and its spectral bandwidth (∆λ). Usually, the spectral bandwidth is described by the full width at half-maximum (FWHM): the width at the intensity level equal to half the maximum intensity. Then, the axial resolution (along the z axis) in free space equals the round-trip coherence length of the source light (Fercher 2010; Popescu et al. 2011; Drexler et al. 2014; Fujimoto and Swanson 2016; Aumann et al. 2019; Brezinski 2006d):

This means, at the theoretical level at least, that a better axial resolution is dependent only on the light source: a shorter wavelength (λ0) and broader spectral bandwidth (∆λ).

The lateral resolution is determined by the spot size of the probing beam, which is dependent on the optics of the imaging device (Fig. 2b; NA = numerical aperture of focusing lens) (Fercher 2010; Popescu et al. 2011; Drexler et al. 2014; Fujimoto and Swanson 2016; Aumann et al. 2019).

Tight focusing would increase the lateral resolution (via the increase of the NA) at the expense of focusing depth (the axial range). The confocal parameter, b, is twice the Rayleigh length (zR):

Tight focusing will reduce the beam waist (ω0: radial size of beam), and the axial range.

Time-domain OCT has several limitations. First, it requires movement of the reference arm corresponding to the z-axis location of the sample arm, which critically limits the scan speed. Second, only the light reflected from a thin tissue slice within the coherence length contributes to the OCT signal, while the detection system records, at each axial location over the full spectral bandwidth, the summated power reflected from all depth locations. To handle this glut of information, Fourier-domain OCT was introduced (Fercher 2010; Popescu et al. 2011; Drexler et al. 2014; Fujimoto and Swanson 2016; Aumann et al. 2019; de Boer et al. 2017a).

4 Fourier-Domain OCT

A Fourier series is an expansion of a periodic function in terms of an infinite sum of sines and cosines. Let us imagine two waves with different frequencies but the same amplitude and phase.

W1(x, t) = A cos (k1x − ω1t) and W2(x, t) = A cos (k2x − ω2t.)

The summation of the two waves is as follows:

This equation clearly shows that the summated wave has two oscillating components: one with a rapidly varying part and the other with a slowly varying envelope (Fig. 3a) (Aumann et al. 2019; Brezinski 2006d). This means that the individual frequency components still exist, even after the summation of the waves. A Fourier series states that every singular frequency component can be traced back from the complex summated waves.

Fig. 3

Working principle of Fourier-domain OCT (FD-OCT). Despite the mixing of diverse frequencies, the summated waves (red waves) still bear the original individual components (a). With formulaic calculation, each component can be extracted from the mixed waves: Fourier-transformation (b; figure taken from Wikimedia). To understand how it works, waves should be described in the complex number system (c; figure taken from Wikipedia). In this polar coordinate system, any point in the waves is described by its real number portion and its imaginary number portion. Then, the waves can be translated into circular movement (phase information is translated into the angle from the x-axis, and the amplitude is translated into the distance from the reference point). Based on the Euler’s equation, any point in the unit circle is cosφ + i sin φ, and can be translated as eiφ (c; figure taken from Wikipedia). Therefore, any wave can be transformed to rotate along the unit circle by simply multiplying eiφ. The frequency profile is squeezed out during the integral process because the hills and valleys of complex waves are counterbalanced during the integration except for when the winding frequencies match exactly those of the individual frequency components. This means that although Fourier-domain OCT receives signals as summated waves, their individual frequency components can be back-calculated and isolated systematically. The total waves are obtained either by a spectrometer (a spectrometer-based OCT: Spectral-domain OCT) or by rapid sweeping of wavelengths from the source (Swept-source OCT). Both implementations record an interference spectrum that carries the depth information of the sample. Fast Fourier Transformation is then used to transform